Evolution under Fertility and Viability Selection

Evolution under Fertility and Viability Selection

Thomas Nagylaki

Department of Molecular Genetics and Cell Biology, The University of Chicago, Chicago, Illinois 60637 Manuscript received June 1 1, 1986

Accepted October 27,. 1986

ABSTRACT

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability

is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in

dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s

[i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is

the fertilities of matings that change slowly [at rate 0 ( s 2 ) ] , the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.

F generations are discrete and nonoverlapping,

I

mating is random, and viabilities are constant, then the mean viability is nondecreasing under viability selection at a single multiallelic locus in a monoecious

population (MULHOLLAND and SMITH 1959; SCHEUER

and MANDEL 1959; ATKINSON, WATTERSON, and

MORAN 1960; KINGMAN 1961). This theorem pro- vides both a conceptual framework for the theory of natural selection and a powerful tool for analyzing the local and global evolution of populations. Therefore, much effort has been devoted to its exact and approx- imate extension.

T h e simplest exact extension is to incorporate mul- tiplicative fertilities. If the fertility of each mating can be expressed as a product of factors corresponding to the genotypes involved, then the model reduces to that for pure viability selection, the viability of each genotype being replaced by the product of its viability and fertility (PENROSE 1949; BODMER 1965; KEMP- THORNEand POLLAK 1970; NACYLAKI 1977a, pp. 51- 55). Consequently, if the genotypic fertilities are con- stant, the mean fitness, naturally defined as the mean viability times the mean fertility, is nondecreasing. For multiple loci, EWENS (1 969a,b) has shown that if fitnesses are additive between loci, the mean fitness is nondecreasing.

Approximate extensions have focused on FISHER’S

( 1930) fundamental theorem of natural selection- that the rate of increase of the mean fitness is equal to the genic variance. T h e work of KIMURA (1958, 1965) and HARTL (1972) suggests that this theorem

Genetics 115: 367-375 (February, 1987)

holds only for weak selection. If several generations have elapsed, the change per generation of the viabil- ities and genotypic fertilities due to their explicit time dependence (if any) is at most second order in the selection intensity s [i.e., 0(s2)], and their total change per generation is smaller than second order [i.e.,

0(s2)], then the theorem holds to O ( s 2 ) for one and

two loci in a continuous-time model of a monoecious population (NAGYLAKI, 1976, 1977a, pp. 79-92, 182- 188). With discrete, nonoverlapping generations, ex- tensions to two loci in a monoecious population (NAGYLAKI 1976, 1977a, pp. 167-177) and to single autosomal and X-linked loci in a dioecious population (NAGYLAKI 1979) depend on the additional assump- tion of multiplicative fertilities.

Since the generalizations described in the last par- agraph are approximate, exceptions occur for every selection pattern. However, the mean fitness can de- crease only if the genic variance is much smaller than s2, which is the case if and only if the absolute value of the gene frequency change per generation is much smaller than s. T h e latter can result from symmetry conditions or proximity to an equilibrium (MORAN 1964; KIMURA 1965; NAGYLAKI 1977b). Stable cy- cling in two-locus models (AKIN 1979, 1982, 1983; HASTINGS 198 1) provides the most striking example of the occasional failure of the fundamental theorem of natural selection.

Much work has been devoted to the analysis of particular models with nonmultiplicative fertilities

therein). Even if there are no viability differences, the fertilities of matings are constant, and generations are discrete and nonoverlapping, the mean fertility can decrease at a single locus in a monoecious population (HADELER and LIBERMAN 1975; POLLAK 1978).

Recently, ABUCOV (1 985) has studied evolution at single diallelic autosomal and X-linked loci in a dioe- cious population under weak selection. He posited discrete, nonoverlapping generations and constancy of the viabilities and of the fertilities of matings, and found that the product of the mean viability and the square root of the mean fertility is nondecreasing. Although his treatment requires some corrections and further analyses, we shall see below that his conclusion is usually correct and can be generalized, proved rigorously, related to the fundamental theorem of natural selection, and illuminated biologically.

In this paper, we shall investigate evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability. T h e fertilities and via- bilities may depend on the genotypic frequencies and time. We shall find nondecreasing quantities under the alternative assumptions that the fertilities of gen- otypes or of matings vary slowly, and we shall deter- mine the rate of increase of these quantities.

Mating is random. We posit discrete, nonoverlap- ping generations for the exploration of dynamics at autosomal and X-linked loci in dioecious populations, but analyze monoecious populations in both discrete and continuous time.

M O N O E C I O U S P O P U L A T I O N

We begin with the case of discrete, nonoverlapping generations and multiplicative fertilities, which is easy to treat exactly and reveals the essential difference between fixing the fertilities of genotypes and fixing those of matings. Next, we analyze arbitrary fertilities. Finally, we show that the same results hold in contin- uous time.

Multiplicative fertilities: The allele A, has fre- quency p , in zygotes. Assume that at least one gener- ation of random mating has occurred and denote the viability of an AzA, individual by vg. Then the fre- quency of ordered A,A, genotypes in adults reads

p z

= pIp]v,/v, 5

2

v4pIpJ. (1)

,

If N , M , j&, and

6,

signify the numbers of adults, matings, and offspring of an A,A, X &Al mating and of an A d J adult, respectively, we have

Instead of

gq,

it will be more convenient to use the scaled genotypic fertility

(3)

If the population is regulated to constancy or is mo- nogamous, and in many other situations, N / M will be constant.

T h e results in this paragraph are established in NACYLAKI (1 977a, pp. 5 1-55). Suppose f ; l , k l may be

expressed as

$j,kl = P i j P k l (4)

p..

9 = b . p ’ / z , ’ (5)

for some Pq. Then

where

designates the mean fertility. The fitness of A,AJ and the mean fitness are given by

wg = u,b,, zlr =

i7.

(7)

Here and below, note carefully that we average via- bilities with respect to zygotic frequencies and fertili- ties with respect to adult frequencies. The frequency of A, in the next generation is

p l

= p , w , / G ( 8 4

wt = wIjpJ, zlr = wt$tpJ. (8b)

where

I II

We assume that the viabilities v,, are constant. On the one hand, if the genotypic fertilities b, are con- stant, so are the fitnesses w,, and hence (8) and the classical theorem stated in the first sentence of the introduction imply that the mean fitness zi, = ??t is

nondecreasing. On the other, by (4), constancy of the fertilities of matings, &l, implies constancy of

P,.

Since (5) enables us to replace w , in (8) by the scaled fitnesses W,, = vz1/3,, now we conclude from (5) and (8) that the modified mean fitness = iP =

177”~

is nondecreasing.

The essential point is that, on account of (3), the fertilities of genotypes and of matings cannot both be constant, and therefore the two different biological assumptions of constancy lead to different results.

General fertilities: We rely on the formulation in NAGYLAKI (1977a, pp. 51-55). Let P , denote the frequency of ordered A,AJ genotypes in zygotes. Equa- tion (1) must be generalized to

p; = p,v,/v, 5 =

c

v p , , (9)

tl

but (2), (3), (6), and (7) still hold. T h e zygotic fre- quencies in the next generation read

Fertility and Viability Selection 369

piwi =

c.

wig+

w

=

1

wijpij. ( 1 1 )

j ij

Introducing the selective differences

rij

= wij

-

w,

l-;

= WI

-

w,

(12)

APi = PJi/G. (13)

we can rewrite (sa) in the form

We now posit weak selection. From ( 1 0) we see that we can scale the viabilities and fertilities by dividing them by the largest viability and the largest fertility, respectively. After scaling, we define the selection intensity s as the greater one of the largest absolute viability difference between genotypes and the largest absolute fertility difference between matings. Thus, we have

vij = 1

+

O(s), Jj,M = 1

+

O(s) ₍₁₄₎

for every

i,

j ,

k

and 1. Throughout this paper, order estimates in s are uniform in the genotypic frequencies and time.

Our first task is to prove that the deviations from Hardy-Weinberg proportions,

R . . 9 = p ,

-

p p .

8 I’ (15)

become small after one generation and nearly con- stant after two. From (10) and (14) we obtain at once

P; = p,pj[l

+

O(s)]. ( 1 6 )

Api = p i o ( s ) ,

(17)

(18) Summing ( 1 6) over j , we infer

whence

P; =

pipj

[ l

+

O(s)].

If t (= 0, 1 ,

.

. .

) designates time in generations, (15) and ( 1 8) reveal

R;j(t) = piPJO(s), t 2 1 . (19)

APq = p;PJO(s), t 3 1 . (20)

Furthermore, ( 1

7)

and (1 8) inform us

In view of ( 1 6) and ( 1

7),

we may write the recursion relation for Re as

Rh = sgij(P, t ) , (21)

ARh SAgij(P, t ) , (22a)

(22b) where gij is bounded as s + 0. Hence, we have

Agij(P, t ) = (gij[P(t

+

I), t

+

11

-

gij[P(t), t

+

111

+(gij[P(t), t

+

13

-

gij[P(t), t31.

A Taylor series and (20) show that the first brace in (22b) is O(s) for t 3 1. Provided the change per

generation of the viabilities and fertilities due to their explicit time dependence (if any) is O(s2), the second brace is also O(s). Then we conclude from (22)

ARij = O(s2), t 3 2. (23)

Before turning to the evolution of the mean fitness, we note that (3),

(7),

and ( 1 4 ) imply

bij = 1

+

O(S), ~ i= j 1

+

O ( S ) , ₍₂₄₎

(25) which reduces ( 1 3) to

Ap; = p i l i

+

O(S*).

This is the more precise version of (1

7).

From (1 1) we find for the change in mean fitness A$ = _{[ ( W e}

+

_Awjj)Ph

-

_w$ij]

ij

~

= AW

+

wjjAPjj, (26)

ij

in which

-

Aw = PhAw,

(27)

ij

represents the change in genotypic fitnesses averaged over the next generation. Since

APq = 0, (28)

ij

we can replace wv in (26) by {U; invoking (15), we get

A 5 =

G

+

2

{ij[piAPj

+

PjApi

(29) ij

+

(Api)(Apj)

+

ARij].

Employing ( 1 7 ) , (23), and (25) in (29) yields

AG =

G

+

2 {ijS;pipj

+

O(S’), t L 2. (30) ij

From ( 1 l ) , (1

2),

( 1 5), and ( 1 9) we deduce

l-i =

p;’

C

lij(pipj

+

Rij) =

C

lijpj

+

O(s2) (31)

j j

for t 3 1; inserting (31) into (30) gives

AG =

v,

+

G

+

0 ( ~ 3 ) ,

v,

= 2

c.

g p i .

t 3 2, (32)

(33) where

i

It remains only to show that for weak selection the variance V , is very close to the genic variance V,. If cy; denotes the average effect of A; on fitness, we have (FISHER 1930; KIMURA 1958; NACYLAKI 1977a, pp. 89-90)

p i < i = p i a i

+

C

Pijaj, (34)

(35)

j

v,

= 2

C

liaipi. i

lead to

Since {I = O(s), from (19) and (36) we obtain

= a ,

+

O(s2), t 5 1. (37)

V , = V,

+

0 ( $ 3 ) , t 3 1, (38)

(39) With the aid of (37), we infer from (33)

and therefore (32) becomes

A G ~ =

v,

+

G

+

0 ( ~ 3 ) , t 3 2.

Consequently, excluding the exceptional cases dis- cussed in the introduction, A 5

=

V, if

G

= o ( s 2 ) . By

(7)

and (27), the average change in the genotypic fitnesses will be o ( s 2 ) if

Aus = o(s2), Ab, = o(s2), (40)

i.e., if the viabilities and genotypic fertilities are nearly constant. Thus, (40) generically suffices for the valid- ity of Fisher's fundamental theorem of natural selec- tion.

Suppose now that the fertilities of matings, rather than those of genotypes, vary slowly: instead of (40), we posit

AV, = O ( s 2 ) , AJj,kl = O(s2). (41)

We proceed to show that the contribution of

G

to (39) is no longer negligible, and its evaluation leads to a different nondecreasing function.

For U,, the analogue of (30) reads

(42)

t 3 2;

the last step comes from (14) and (41). Appealing to (3), (41), the fact that (28) also holds for P ; , (14), and (20), we find

Ab, = C j i , k l A p ;

+

O(s2) (43)

= O ( s 2 ) , t 5 1. (44)

A b =

2

P:Ab,, (45)

A b =

C

$j,kIp;ApG O ( s 2 ) (46)

= O(s2), t 3 1. (47)

kl

With the definition -

?I

(43) and (44) imply -

q k l

From (6), (41), (45), (46), and (47) we derive

= O(S*), t 3 1. (49)

The following simple result will be useful here and in the subsequent sections. Let h =

4y$*,

where y and 6 denote constants and the functions

4

and )I satisfy

4

= 1

+

O(s), $ = 1

+

O ( s ) , (504

A 4 = O(s2), A$ = O ( s 2 ) . (50b)

Then Taylor's theorem gives

ah dh

d 4

a*

Ah = - A 4

+

-

A$

+

O [ ( A b ) *

+

(A$)']

=

+

SAG

+

0 ( ~ 3 ) . (51)

AG = AV

+

A T + O ( s 3 ) , t k 2 . (52)

(53) Recalling

(7),

(14), ( 4 2 ) , and (49), from (51) we get

Next, we invoke

(7)

and (41):

Aw, = u,Ab,

+

o(s2).

We can calculate

G

by successive use of (27), (53), (20), (44), (9), (45), (14), (44) again, and (48):

aw

=

2

P$u,Ab,

+

o(s2) y

= P,u,Ab,

+

o ( s 2 ) , t 2 1

= P$GAb,

+

o(s2), t 2 1

Ij

s -

= Ab

+

o ( s 2 ) , t k l

= !hAT+ o(s2) t 2 1. (54)

(55)

A m = V,

+

o ( s 2 ) , t 3

2 ,

(56) Substituting (52) and (54) into (39), we obtain

A(V

+

!h

7)

= V,

+

o(s2), t k

2 .

With the aid of (51), we can transform (55) to

in which =

Gy1"

signifies the modified mean fitness. Consequently, as in the special case of multiplicative fertilities, here

@

is nondecreasing.

Continuous time: Here we prove that, mutatis mu-

tandis, the results for discrete, nonoverlapping gen- erations remain valid for a continuous-time model with no age dependence or a stationary age distribu- tion. Let dq,Jj,kl, and b, designate the mortality rate of an AiAj individual and the fertilities per unit time of an AiA, X AkAl mating and an AiA, individual, respectively. The mean fertility and the mean mortal- ity read

7

= jj,klP$ki,

H

=

2

d . P . . ?I 8' (57)

Fertility and Viability Selection

where P , represents the frequency of ordered A,A,

genotypes at time t; we define the fitness of

AAj

and the mean fitness as

(58)

m.. _rj = b.. _ZJ

-

d,, fi =

7

-

z.

Suppose now

in which d and b denote positive constants, and

If the gene frequencies are bounded away from zero, for which it is sufficient but not necessary to assume that t c/s for some constant c, then (NAGYLAKI

1976, 1977a, pp. 79-92)

A

=

v,

+

m'

+

o(?),

_t 5 t 2 , (61) where t 2

=

-2b-' In s, the superior dot signifies the time derivative, and

-

(62)

m

=

mp..

1 1J'

ij

If the mortalities and genotypic fertilities vary slowly, i.e.,

ciij = o(s2), dij = o(s2), (63) then (61) implies that

4

=

V,, so f i is generally non- decreasing.

If the fertilities of matings are nearly constant, instead of (63) we assume

(64)

(65)

djj = o(s2), Jj& = o(s2).

From (57) and (64) we deduce

f

=

2

2

f;j.,klpijpkl

+

o(s2), ijkl

-

d

= Z f j , k l P $ k l

+

O(S2) = %?

f

+

O ( S 2 ) . ₍₆₆₎ ijkl

Substituting (58) and (66) into (64) leads to

-r

M = V,

+

o(s'), t 5 t 2 , (67)

where the modified mean fitness is

ifi

= $4

7

-

z.

analogues of (39) and (56), respectively.

Clearly, (61) and (67) are the precise continuous

AN AUTOSOMAL LOCUS IN A DIOECIOUS POPULATION

For simplicity, we assume that the locus under con- sideration does not influence the sex ratio at birth. Then the genotypic frequencies are the same in male and female zygotes. Let Pij and

pi

denote the frequen- cies of ordered A,Aj genotypes and Ai alleles in zygotes, respectively. Single and double asterisks signify the respective frequencies in adult males and females. If

u y and uij designate the viabilities of AiA, males and

females, respectively, we have

P; = P,u,/U, U = UaJ'rj, (684 rj

P;* = py7J,/v,

v

=

vzprj.

(68b)

'I

Let J j , k l represent the number of progeny from the

mating of an

AA,

male with an A d l female. We do not assume that reciprocal crosses are equally fertile,

i.e., we allowJ;,,kl Z ~ L , , ~ . T h e zygotic frequencies in the next generation are

1

p; =

-

(PaP$*$k,!,

+

p$f%*Aj,zk), (69)

2f kl

where the mean fertility reads

Let M, N* and N** denote the numbers of matings and adult males and females, respectively. If (zij and designate the respective fertilities of A;Aj males and females, we get the scaled fertilities

a.. 'J = N*ii../M rj = cAj,klP$*,

( 7 1 4

b, = N**&/M = cfkl,$$, (71b)

rj rj rj

7.

(72)

k1

kl

whence we obtain the mean fertilities

5 = ai?* = f,

b=

2

j..P** =

ij ij

Define the fitnesses of A,Aj males and females as

(73)

x . . = u - . a . . _{g rj aJ?} y . . = v . . b . . _lj

IJ rj-

Then (73), (68), and (72) give the mean fitnesses of males and females:

-

-

(74)

x

= x ; p i j = U f, j =

yip..

= ; f.

'I

Ij ij

T h e fitnesses xi and y; of Ai in males and females, respectively, are given by

(75)

p;xi = x#?.

p .

. = y . . p . . t '13 zJa 'I ?J'

j j

We sum (69) o v e r j and use

(71),

(68), (73), (74), and (75):

We scale so that

(77)

uij = 1

+

O(s), vij = 1

+

O(s),

$j,kl = 1

+

o(s),

wtj = %(Xq

+

ye) (78)

as the average fitness of A A , and retain (1 1) and (1 2). Inserting (78) into (1 1) produces

G =

%(X

+

j ) , (79)

and using

(77),

(78), (79), and (12) in (76) establishes (25). As in the monoecious case, (39) holds.

If the viabilities and genotypic fertilities are almost constant, a.e.,

A u , = o(s'), AV, = o(s2), ( 8 0 4

A a , = o(s2), Ab, = o(s'), (80b)

then (39) informs us that Arl,

=

V,, so the mean fitness generally increases at a rate equal to the genic vari- ance. We can prove the same result for the geometric mean fitness. By direct analogy with (42), we have

A? = O(s2), A y = O(s') (81)

for t 3 2. Therefore, we may apply (51), so that (39) and (79) demonstrate

A; = V,

+

o(s'), t Z 2, (82) where = ( 2 j ) ' / ' . On account of (74), we can express

6

as Zj =

U T ,

in which

0

= (ii5)"' represents the geometric mean viability.

Now let us retain (80a), but posit that the fertilities of matings are almost constant; thus, we replace (80b) by

AJ],kl = o ( s 2 ) . (83)

Aii = O ( s 2 ) , A 5 = O(s'), t Z 2; (84) As in (42),

as in (43) and (44),

A a , = c j j , k $ ; *

+

O ( S 2 ) O($'), t 2 1, (85a)

kl

Ab, = cfkl,,y$

+

o ( s 2 ) = O ( s 2 ) , t 2 1. (85b)

kl

instead of (48), now we have

A T =

aa

+

ab

+

o ( s ' ) , t 3 1, (86) where

-

__

A a =

2

PETAa,, A b = P:"Ab,; (87)

v 'I

(49) still holds. From (79), (74), and (5 1) we obtain

AG = %(A;

+

A;)

+

A T + O ( s 3 ) , t 3 2. ₍₈₈₎

As in the derivation of (54), we find

-

Ax =

2

PhAx, =

&

+

o(s2),

Ay =

2

PhAy, = A b

+

o ( s 2 ) ,

t 3 1,

t 2 1.

(89a)

(89b) II

- -

4

Substituting (88), (89), and (86) into (39) yields

%A(;

+

3

+

7)

= V,

+

o(s2), t 3 2. ₍₉₀₎

By (51), we can transform (90) to (56); now the mod- ified mean fitness is the geometric mean of the modified mean fitnesses of males and females:

= (V*W**)1/2

m*

= ; T I / ' ,

,

(91)

E**

=

;p*

In precise analogy with the monoecious result, we have

w

=

Dy'/'.

AN X-LINKED LOCUS

We assume that the locus under consideration does not influence the sex ratio at birth. Let p i ,

Qc,

and q i

denote the frequencies of Ai male zygotes, ordered

A A j female zygotes, and Ai in female zygotes, respec- tively. If ui and v v designate the respective viabilities of A , males and AiAj females, we have the adult fre- quencies

pT

= p$L,/ii,

ii

=

c

U&, ( 9 2 4

Q* J = Q . v . . / y , 1 4 5 = ( 9 3 4

I

ij

Letf,g represent the number of progeny from the mating of an Ai male with an female. T h e zygotic frequencies in the next generation are

1

f

kl

1

pl

= =

c

pk*Q;ji,,l, ( 9 3 4

QI$

=

--

1

(fi?

@?J,kj

+

pj*Qa%h,ik),

(93b)

2f

k

where the mean fertility reads

T =

c f ; , j k p ? Q $ . (94) ijk

We define M , N* and N** as in the previous section and let a"i and

6v

designate the respective fertilities of

Ai males and AiAj females. Then we get the scaled fertilities

= N * & / M =

f l , k d $ ,

( 9 5 4

(95b)

k l

6, = N**&,/M =

2fk,epff,

k

whence

-

6 =

2

a@? = f ,

b

=

2

b-Q* 9 , =

7.

₍₉₆₎

I 0

With fitnesses

(97) x. a = u.a. t a j y.. g = v..b. i ] t j

Fertility and Viability Selection 373

x

=

2

x.p. I t =

67,

j

=

1

yqQy =

777.

(98)

i ij

We complete our formulation by using (93), (95), (92), (97), and (98):

p(

= q .

./-,

= - 1

(iE

+

y),

(99)

2 2

9 1 Y

where

q y . Z E =

1

)I..(&. 'tl I' (100)

j

T h e analysis of the dynamics is a little more involved than in the autosomal case, and the approach to the neighborhood of the Hardy-Weinberg surface is not quite as rapid. We scale so that

uj = 1

+

O(s), vy = 1

+

O(s),

(101)

j , j k = 1

+

O(s),

and set

Pi =

%(pi

+

2qi),

~i =

Rq = Qj

-

P f j , (102)

(103)

Rq = qi

-

Pi = % ( q i

-

pi);

j

Pi signifies the mean frequency of A, in zygotes, and

R , is a measure of deviation from Hardy-Weinberg proportions. From (1 02) and (1 03) we have

pi

= Pi

-

2ri, qi = Pi

+

ri, Qq = PiPj

+

R+ (104) Let

t a = x i - x ,

q . . = y . . - j 1 3 ' 1 , q . = y i - y . t

-

(105) Appealing to (102), (99), (101), and (105), we find

APi =

%(piti

+

2 q ~ i )

+

O(s2) (1 06)

= O(s). (107)

Equations (92), (93), and (101) yield

p(

= qi

+

O ( S ) , QG = 1/2(piqj

+

pjqi)

+

O ( S ) , (108) whence

q( =

Yz(p;

+

qi)

+

O ( s ) .

r: = -%Ti

+

O ( s ) .

(109)

(110)

Iri(O)l(l/2)11 6 s (111)

r i ( t ) = O(s), t 3 t l . (1 12)

From (1 03), (1 OS), and (109) we infer

If t l is the shortest time such that

for every

i,

iterating (1 10) establishes

Invoking (102), (108), (107), (104), and (1 12) leads to

RG = O(s) f o r t 3 t l , which means

Rq(t) = O(s), t

>

t l . (113)

From (1 04), (1 07), and (1 12) we derive

A& = O ( S ) , t 3 t i , (114a) and (104), (1 07), and (1 13) demonstrate

AQj = O ( S ) , t > t i . (114b) T h e proof of (23) now gives

ARy = O ( s 2 > , t 2 t~ = t i

+

2, ₍₁₁₅₎ provided the rate of change of the viabilities and fertilities due to their explicit time dependence (if any) is O(s2).

Before proceeding to the evolution of the mean fitness, we employ (1 04) and (1 12) to rewrite (106) in the form

APi = Pili

+

O ( s 2 ) , t 3 t i , (116)

(117) where

li

=

%(ti

+

27;).

By manipulations similar to those that produce (29),

A2 =

z

+

t i A f i i , Ay =

+

qyAQ,, (1 18)

from (98) and (105) we find

i g

where

-

-

A X =

p(

AX;, Ay = Q4Ayq. (1 19)

i ij

Despite the unequal weighting in (1 17), we must weight the male and female mean fitnesses equally (HARTL 1972; NAGYLAKI 1979). Therefore, we retain (79), but now define

X

and j by (98). From (79) and (1 18) we obtain

AG =

KG

+

!h EjAfij

+

!h qqAQj, (120)

i y

in which

-

Aw =

?h(z

+

ay).

₍₁₂₁₎

Applying (1 07) and (1 15) to (104), we see that

Api = APi

+

O(S'), t b t 2 , (122a)

AQj = PiAPj

+

PjAPi

+

O(s2), t 3 t2. (122b)

Suppose now that the gene frequencies P, are bounded away from zero. For this it is sufficient but not nec- essary to assume that t G c/s for some constant c.

Then, by (105), (104), (1 12), and (1 13),

qi = qijpj

+

O(s2), t

>

ti. (123)

j

Substituting (1 22), (1 23), (1 16), and (1 17) into (1 20) produces the analogue of (32):

(124)

AG = V ,

+

aw

+

O ( s 3 ) , t b t z , where

Equations (33) and (1 25) both display the mean num- ber of genes (2 and 3 / ~ , respectively) at the locus under

consideration.

T o establish the analogue of (39), we must prove that V , is close to the genic variance V,. We analyze the total variance

v

= !43

2

g p i

+

'/3

2

72.Q.. Y Y ₍₁₂₆₎ i i j

with respect to deviations from complete dosage com- pensation and no dominance

( 4

special cases in HARTL 1972; NACYLAKI 1979):

E;

= 2ai

+

di, 7ij = ai

+

aj

+

Dij. ₍₁₂₇₎

T h e weighting in (126) is by the number of sex chromosomes per individual, as in (1 02) and (1 17); ai,

di, and

D,

represent the average effect of Ai on fitness and the dominance deviations from complete dosage compensation in males and females, respectively. T h e genic and dominance variances read

(128a)

(1 28b)

Fixing

ti

and 7 y and minimizing V d with respect to V, = $4

Vd =

'/s

2

pid?

+

'%

2

QjD$

pj(2ai)'

+

?A

2 Qj(ai

+

aj)',

i ij

i rj

a , leads to

pi([i

-

2ai)

+

2

2

Qj(7ij

-

( ~ i

-

aj) = 0; (129)

j

in view of (1 27), this is identical to

p;di

+

2 QjDy = 0. (1 30)

j

With the aid of ( 1 29), we can reduce (1 28a) to

V, =

?A

2

ai(piti

+

2qi~i). (131)

i

Summing (129) over

i

and recalling (105) and (102) proves that the mean additive effect is zero:

2

ajp; = 0. (132)

i

We insert (1 27), (1 28), and (1 30) into (1 26) to dem- onstrate the additivity of the variance components:

v

=

v,

+

V d . ₍₁₃₃₎

Successive use of (104), (132), (117), (112), and (1 13) in (1 29) yields

ai = %(j

+

O(s'), t

>

t l . (134)

Employing (1 04), (1 12), (1 17), (1 34), and (1 25) sim- plifies ( 1 3 1) to

v,

=

v,

+

O(s3), t > t l , (135)

where (1 24) becomes

Azzl = V,

+

nw

+

O(s3), t 3 t Z . ₍₁₃₆₎

If the viabilities and genotypic fertilities are nearly

A u , = o(s'), AV, = o(s'), ( 1 37a)

Aa, = o(s'), Ab, = o(s'), (137b)

constant, i.e.,

then (1 36) becomes

Azzl = V,

+

o(s'), t 2 t z , (138) so after a short time t ~ , Zi, generally increases. To see that (138) also holds for & = ( i j ) ' / ' , we observe that (1 IS), (1 37), (1 O l ) , and (1 14) imply the validity of (81) for t

>

t l . Hence, (51) applies, and the desired result follows from (79) and (138). By (98), we have & =

07,

where

0

=

(U

ij)l/', as in the autosomal case.

Now we retain the hypothesis (1 37a) for the viabil- ities, but instead of (1 37b) posit approximate con- stancy of the fertilities of matings:

= o(s2). (1 39)

Writing (118) for zi and ij and appealing to (137a), (1 0 l), and (1 14) demonstrates

Aii = O(s2), A 5 = O(S'), t

>

t i . (140) From (95), (139), (92), (137a), (101), and (114) we obtain

(141a) Aa, = c j , u A Q j , i

+

o(s') = O(s'), t

>

t l ,

kl

Ab, = Cfk,,Apf

+

o(s') = O(S'), t 2 ti. (141b) k

We invoke (94), (139), (141), (92), (101), and (114) to prove

J =

z

+

ab

+

o(s'), t

>

t l , (142)

where

- -

A a = p,*Aa,, A b = @Ab,. (143)

t B

On account of (141) and (142), we have

A Y = O(s'), t

>

t l . (144) From (79), (98), and (51) we get

AG = %(A;

+

A 5 )

+

A Y + O(S'), t

>

t l . ₍₁₄₅₎

Furthermore, (97), (137a), (1 19), (1 14), (92), (141), and (1 43) imply

A x =

aa

+

o ( s 2 ) , t 2 t l , (1 46a)

Ay =

ab

+

o(s'), t

>

t l . (1 46b)

Inserting (145), (121), (146), and (142) into (136) yields (90) for t 2 t 2 . With the definition (91), we

deduce

-

-

A F = V,

+

o(s'), t 2 t 2 (147) for the modified mean fitness

autosomal locus.

Fertility and Viability Selection

29). Clearly, in this case the first assumption makes

DISCUSSION

We have investigated the dynamics at a single mul- tiallelic locus under arbitrary weak selection on both fertility and viability. We posited discrete, nonoverlap- ping generations for autosomal and X-linked loci in dioecious populations, but studied monoecious popu- lations in both discrete and continuous time. Mating is random in all cases. T h e results hold after several generations have elapsed. T h e divergence of the pop- ulation from Hardy-Weinberg proportions is O(s), where s represents the selection intensity. Provided the change per generation of the fertilities and viabil- ities due to their explicit time dependence (if any) is

O ( s 2 ) , the rate of change of the deviation from Hardy- Weinberg proportions is O ( s 2 ) .

T h e mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. As shown in (39), (61), and (136), the rate of change of the mean fitness (AG) is close to the sum of the additive component of the genetic variance (V,) and the average rate of change of the genotypic fitnesses

(z),

the error being O ( s 3 ) .

Suppose the change per generation of the viabilities is o ( s 2 ) . Then different biological assumptions con- cerning the approximate constancy of the fertilities crucially affect the value o f z , and thereby influence the form of the Fundamental Theorem of Natural Selection.

On the one hand, if the change p e s e n e r a t i o n of the fertilities ofgenotypes is o ( s 2 ) , then Aw is negligible, and hence AG

=

V,, as stated below (39) and (63) and in (82) and (1 38). This implies that as long as there is substantial gene frequency change, the mean fitness increases.

On the other hand, if it is the fertilities of matings

that change at a rate o(s2), then

z

contributes signif- icantly and now A m

=

V,, where the modified mean fitness is the product of the mean viability and the square root of the mean fertility; see (56), (67), below (go), and (147).

T h e choice between the alternative assumptions discussed in the last two paragraphs can only be made on biological grounds; both assumptions lead to sen- sible mathematical models. Perhaps the assumption of nearly constant genotypic fertilities is reasonable for polygamous species, whereas the fertilities of matings may be almost constant in monogamous ones. It is easy to see that if, instead of mating, each genotype sheds gametes and these gametes fuse at random, then the standard selection models with zygotes in gener- alized Hardy-Weinberg proportions apply, and each genotypic fitness is the product of the viability and half the number of gametes produced

(6

CROW and KIMURA 1970, pp. 188-189; NAGYLAKI 1977a, pp. 54,155-156,163-164; ROUGHCARDEN 1979, pp. 26-

biological sense, whereas the second has no meaning.

I thank ETHAN AKIN for helpful comments. This work was supported by National Science Foundation grant BSR-85 12844.

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